ECE 595, Section 10 Numerical Simulations

Lecture 10: Solving Quantum Wavefunctions

Prof. Peter Bermel January 30, 2013

Outline

• Recap from Monday • Schrodinger’s equation • Infinite & Finite Quantum Wells • Kronig-Penney model • Numerical solutions:

– Real space – Fourier space

1/30/2013 ECE 595, Prof. Bermel

Recap from Monday

• Application Examples – Electrostatic potential

(Poisson’s equation) • 1D array of charge • 2D grid of charge

– Arrays of interacting spins • 1D interaction along a chain • 2D nearest-neighbor coupling

1/30/2013 ECE 595, Prof. Bermel

Electrostatic potential in 2D

(7x7 grid)

Schrodinger’s Equation

• Wavefunction Ψ describes extent of particle • Also eigenfunction of Schrodinger’s equation:

ℋΨ = 𝐸Ψ • Hamiltonian consists of kinetic and potential

terms: ℋ = 𝑇 + 𝑉

• Classically, 𝑇 = 𝑝2

2𝑚; if 𝑝 = −𝑖ℏ𝛻, 𝑇 = − ℏ2

2𝑚𝛻2

• Probability of finding at x given by Ψ(𝑥) 2

1/30/2013 ECE 595, Prof. Bermel

Free Particle • A free particle has zero potential

everywhere • Schrodinger’s equation becomes:

−ℏ2

2𝑚𝛻2Ψ = 𝐸Ψ

• Eigenfunction can be obtained analytically:

Ψ 𝑥 = 𝐴𝑒±𝑖𝑖𝑖 • Energy eigenvalue thus given by:

𝐸 =ℏ2𝑘2

2𝑚

1/30/2013 ECE 595, Prof. Bermel

Infinite Quantum Well

• Example: proton in iron nucleus

• Potential 𝑉 𝑥 = �0, 𝑥 < 𝑎/2∞, |𝑥| ≥ 𝑎/2

• Boundary condition: Ψ ±𝑎/2 = 0

• Eigenfunctions are standing waves: Ψ 𝑥 = 𝐴 𝑒𝑖𝑖𝑖 + 𝑒−𝑖𝑖𝑖

• By BC’s, 𝑘 = 𝑛𝜋𝑎

; 𝐸 = ℏ2𝑛2𝜋2

2𝑚𝑎2

1/30/2013 ECE 595, Prof. Bermel

a

Finite Quantum Well

• Example: α-particle in U-235 nucleus

• Potential 𝑉 𝑥 = �0, 𝑥 < 𝑎/2𝑈, |𝑥| ≥ 𝑎/2

• Boundary conditions: Ψ ±(𝑎 + 𝜖)/2 = Ψ ±(𝑎 − 𝜖)/2 Ψ′ ±(𝑎 + 𝜖)/2 = Ψ′ ±(𝑎 − 𝜖)/2

• Eigenfunctions inside box like before; outside region decays exponentially

1/30/2013 ECE 595, Prof. Bermel

−𝑎/2 𝑎/2

Kronig-Penney Potential

• Example: 1D atomic crystal • Potential

𝑉 𝑥 = �0, 0 < 𝑥 < 𝑎/2𝑈, 𝑎/2 < 𝑥 < 𝑎

• And, 𝑉 𝑥 + 𝑎 = 𝑉(𝑥) • Boundary conditions:

Ψ 𝑥 + 𝑎 = Ψ 𝑥 ? • Will each electron be stuck

in its own little well?

1/30/2013 ECE 595, Prof. Bermel

-a -a/2 0 a/2 a

V(x)

x

U

I II I´ II´

Bloch Theorem

“When I started to think about it, I felt that the main problem was to explain how the electrons could sneak by all the ions in a metal …. By straight Fourier analysis, I found to my delight that the wave differed from the plane wave of free electron only by a periodic modulation.” --Felix Bloch, Physics Today (1976)

1/30/2013 ECE 595, Prof. Bermel

Bloch Theorem

• Asserts that solution in periodic potential is always a product of two terms: – a periodic function (with the same period) – a plane wave

• Mathematically, we can write: Ψ 𝑥 = 𝐴𝑒𝑖𝑖𝑖𝑢(𝑥)

where 𝑢 𝑥 + 𝑎 = 𝑢(𝑥)

1/30/2013 ECE 595, Prof. Bermel

Bloch Theorem: Numerical Solution

• Use Bloch’s theorem to solve the eigenproblem numerically

−ℏ2

2𝑚𝛻2 + 𝑉(𝑥) 𝑒𝑖𝑖𝑖𝑢(𝑥) = 𝐸(𝑘)𝑒𝑖𝑖𝑖𝑢(𝑥)

• What basis to use for periodic function?

1/30/2013 ECE 595, Prof. Bermel

Bloch Theorem: Real-Space Basis

• Real space is most obvious, with uniform grid • Pull out plane wave from eigenvector to

reduce complexity:

−ℏ2

2𝑚𝛻2 + 𝑉(𝑥) 𝑢(𝑥) = 𝐸 𝑘 −

ℏ2𝑘2

2𝑚 𝑢(𝑥)

• Immediate problem: not positive definite

1/30/2013 ECE 595, Prof. Bermel

Bloch Theorem: Real-Space Basis

1/30/2013 ECE 595, Prof. Bermel

Bloch Theorem: Fourier Space Basis

• If we write periodic function as Fourier series:

𝑢 𝑟 = �𝑐𝐺𝑒𝑖𝐺𝑖𝐺

• We obtain the nice recursion relation:

𝑉𝐺´𝑐𝐺−𝐺𝐺 = 𝐸 𝑘 −ℏ2

2𝑚𝑘 + 𝐺 2 𝑐𝐺

1/30/2013 ECE 595, Prof. Bermel

Bloch Theorem: Fourier Space Basis

1/30/2013 ECE 595, Prof. Bermel

Physical Observation

From C. Kittel, Introduction to Solid State Physics

1/30/2013 ECE 595, Prof. Bermel

Next Class • Is on Friday, Feb. 1 • Will discuss numerical tools for Fast

Fourier Transforms • Recommended reading: Numerical

Recipes, Chapter 12

1/30/2013 ECE 595, Prof. Bermel